Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Average Error: 15.5 → 3.0

Time: 6.2s

Precision: binary64

\[[x, y] = \mathsf{sort}([x, y]) \\]

Math

\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}\\ t_1 := \left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{t_0 \cdot t_0}\right) \cdot \frac{\frac{\sqrt[3]{y}}{t_0}}{z}\\ \mathbf{if}\;x \cdot y \leq -1.3312952842167844 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 2.2270002756079324 \cdot 10^{-147}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (cbrt (fma z z z)))
        (t_1
         (*
          (* x (/ (* (cbrt y) (cbrt y)) (* t_0 t_0)))
          (/ (/ (cbrt y) t_0) z))))
   (if (<= (* x y) -1.3312952842167844e-99)
     t_1
     (if (<= (* x y) 2.2270002756079324e-147)
       (* (/ y z) (/ x (fma z z z)))
       t_1))))

C

double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

↓

double code(double x, double y, double z) {
	double t_0 = cbrt(fma(z, z, z));
	double t_1 = (x * ((cbrt(y) * cbrt(y)) / (t_0 * t_0))) * ((cbrt(y) / t_0) / z);
	double tmp;
	if ((x * y) <= -1.3312952842167844e-99) {
		tmp = t_1;
	} else if ((x * y) <= 2.2270002756079324e-147) {
		tmp = (y / z) * (x / fma(z, z, z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

↓

function code(x, y, z)
	t_0 = cbrt(fma(z, z, z))
	t_1 = Float64(Float64(x * Float64(Float64(cbrt(y) * cbrt(y)) / Float64(t_0 * t_0))) * Float64(Float64(cbrt(y) / t_0) / z))
	tmp = 0.0
	if (Float64(x * y) <= -1.3312952842167844e-99)
		tmp = t_1;
	elseif (Float64(x * y) <= 2.2270002756079324e-147)
		tmp = Float64(Float64(y / z) * Float64(x / fma(z, z, z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[Power[N[(z * z + z), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(N[(N[Power[y, 1/3], $MachinePrecision] * N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[y, 1/3], $MachinePrecision] / t$95$0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1.3312952842167844e-99], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2.2270002756079324e-147], N[(N[(y / z), $MachinePrecision] * N[(x / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.5
Target	4.4
Herbie	3.0

\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.33129528421678439e-99 or 2.2270002756079324e-147 < (*.f64 x y)

   1. Initial program 14.0

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

   2. Simplified 10.4

      \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}{z}} \]

   3. Applied *-un-lft-identity_binary64 10.4

      \[\leadsto x \cdot \frac{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}{\color{blue}{1 \cdot z}} \]

   4. Applied add-cube-cbrt_binary64 10.9

      \[\leadsto x \cdot \frac{\frac{y}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(z, z, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}}{1 \cdot z} \]

   5. Applied add-cube-cbrt_binary64 11.0

      \[\leadsto x \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\left(\sqrt[3]{\mathsf{fma}\left(z, z, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}{1 \cdot z} \]

   6. Applied times-frac_binary64 11.0

      \[\leadsto x \cdot \frac{\color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}}{1 \cdot z} \]

   7. Applied times-frac_binary64 11.0

      \[\leadsto x \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}{1} \cdot \frac{\frac{\sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}{z}\right)} \]

   8. Applied associate-*r*_binary64 4.6

      \[\leadsto \color{blue}{\left(x \cdot \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}{1}\right) \cdot \frac{\frac{\sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}{z}} \]

   9. Simplified 4.6

      \[\leadsto \color{blue}{\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}\right)} \cdot \frac{\frac{\sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}{z} \]

   if -1.33129528421678439e-99 < (*.f64 x y) < 2.2270002756079324e-147

   1. Initial program 17.5

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

   2. Simplified 7.6

      \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}{z}} \]

   3. Applied associate-*r/_binary64 2.3

      \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}}{z}} \]

   4. Applied associate-/l*_binary64 7.2

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}}} \]

   5. Applied associate-/r/_binary64 7.2

      \[\leadsto \frac{x}{\color{blue}{\frac{z}{y} \cdot \mathsf{fma}\left(z, z, z\right)}} \]

   6. Applied *-un-lft-identity_binary64 7.2

      \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{z}{y} \cdot \mathsf{fma}\left(z, z, z\right)} \]

   7. Applied times-frac_binary64 0.9

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{y}} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}} \]

   8. Simplified 0.8

      \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.3312952842167844 \cdot 10^{-99}:\\ \;\;\;\;\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}\right) \cdot \frac{\frac{\sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}{z}\\ \mathbf{elif}\;x \cdot y \leq 2.2270002756079324 \cdot 10^{-147}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}\right) \cdot \frac{\frac{\sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))